In order for a vacuum character to be identifiable with a 4d Schur index, one must be able to interpret its Fourier coefficients as counting Schur operators. Superconformal representation theory imposes certain constraints.
In Table REF  we list the 4d superconformal multiplets that contain Schur operators (one for each multiplet). It will be useful to recall the interpretation of a few of them (see e.g. [1]}, [2]} for more details):
